Below it, it read "Given the initial formation of ten coins, move exactly # coins to produce the end formation." It was pretty obvious that # stood for a digit, but it was smudged and couldn't be read. What possible numbers could it have been so the problem was solvable?

The question asks more than just what is the minimum; the problem asks
for which numbers could go into # and the question be answerable. For a
given candidate of #, we want to be able to freeze 10-# coins that lie
in some overlap of the "from" and "to" pics.
I find it fairly easy to see many arrangements of the pictures to get
from 0 to 4 overlapping coins, corresponding to # being 10 through 6.
There are also several straight forward ways of getting exactly 6
overlapping coins and the previous posters point out that when the
diagrams are centered over each other, they have 7 overlapping coins.
These correspond to # equal 4 and 3 respectively. I am convinced by an
maximal area argument (but not a proof) that 7 overlapping coins is the
maximum, so 3 is the minimum value of #.
What I don't see is how to get exactly 5 overlapping coins. I suppose
we could pick 5 of a 6-coin overlap, but this seems a cheat to move a
coin from a spot, only to replace it with another. I am stuck on this;
perhaps #=5 is not possible.
A good added condition is given by Brianjn; all moved coins must remain
in constant contact with at least one other coin. Can all of the above
values of # be realized under this constraint?